Example of a ring extension $A\subseteq B$ and $F:A\to \Omega$ such that $f$ cannot be extended to $B$.

The issue is that some elements in $B$ satisfy equations over $A$ and those equations transform into incompatible equations by the map $A\to \Omega$. Here is a typical example

$$A = \mathbb{C}[x] \to \mathbb{C}[x,y]/(x y -1)=B$$

and take the map $A \to \Omega= \mathbb{C}$, $x\mapsto 0$. The equation $xy-1$ satisfied by $y$ over $\mathbb{C}[x]$ is transformed to an impossible equation $0\cdot \bar y -1 =0$.

One issue : some elements of $A$ getting mapped to $0$ in $\Omega$. For instance if the map $A \to \Omega$ is injective and the extension $A\to B$ is finitely generated then we can always extend it to a map $B\to \Omega$ (not necessarily injective). There is a more general result due to Chevalley on extensions of maps to an algebraically close field for finitely generated extensions, should be in the same book.

Another issue ( @Slade: thanks!) is that the extension $A\to B$ may not be finitely generated. Example $A= \mathbb{C} \to \mathbb{C}(x)=B$, cannot extend the map $A \to \mathbb{C}$ to $B$.